Ccs s Math April 2013

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California Department of Education i |

March 2013

Common Core

State Standards

for Mathematicsfor California Public SchoolsKindergarten Through Grade Twelve

Adopted by theCalifornia

State Board of Education

August 2010

Updated January 2013

Prepublication Version


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Prepublication Version, April 2013

California Department of Education

Contents

A Message from the State Board of Education and the State Superintendent of Public Instruction ....... iIntroduction .............................................................................................................................................. ii

Standards for Mathematical Practice ....................................................................................................... 1

K8 Standards ........................................................................................................................................... 5

Kindergarten ................................................................................................................................. 6

Grade 1 ....................................................................................................................................... 10

Grade 2 ....................................................................................................................................... 15

Grade 3 ....................................................................................................................................... 20

Grade 4 ....................................................................................................................................... 26

Grade 5 ....................................................................................................................................... 32

Grade 6 ....................................................................................................................................... 38

Grade 7 ....................................................................................................................................... 45Grade 8 ....................................................................................................................................... 51

Higher Mathematics Standards .............................................................................................................. 57

Higher Mathematics Courses ..................................................................................................... 60

Traditional Pathway ........................................................................................................ 60

Algebra I .............................................................................................................. 61

Geometry ............................................................................................................ 70

Algebra II ............................................................................................................. 78

Integrated Pathway ........................................................................................................ 87

Mathematics I ..................................................................................................... 88

Mathematics II .................................................................................................... 97Mathematics III ................................................................................................. 107

Advanced Mathematics ................................................................................................ 116

Advanced Placement Probability and Statistics Standards ............................. 117

Calculus Standards ............................................................................................ 119

Higher Mathematics Standards by Conceptual Category ........................................................ 122

Number and Quantity ................................................................................................... 123

Algebra .......................................................................................................................... 126

Functions ....................................................................................................................... 130

Modeling ....................................................................................................................... 134

Geometry ...................................................................................................................... 136

Statistics and Probability .............................................................................................. 141

Glossary................................................................................................................................................. 145
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California Department of Education i |

A Message from the State Board of Education

and the State Superintendent of Public

Instruction

A Message from the State Board of Education and State Superintendent of Public Instruction:

The Common Core State Standards for Mathematics (CCSSM) reflect the importance of focus,

coherence, and rigor as the guiding principles for mathematics instruction and learning. Californias

implementation of the CCSSM demonstrates a commitment to providing a world-class education for

all students that supports college and career readiness and the knowledge and skills necessary to fully

participate in the 21st century global economy.

The CCSSM build on Californias standards-based educational system in which curriculum, instruction,

professional learning, assessment, and accountability are aligned to support student attainment of

the standards. The CCSSM incorporate current research and input from other education stakeholders

including other state departments of education, scholars, professional organizations, teachers and

other educators, parents, and students. A number of California-specific additions to the standards

(identified in bolded text and followed by the CA state acronym) were incorporated in an effort to

retain the consistency and precision of our past standards. The CCSSM are internationally

benchmarked, research-based, and unequivocally rigorous.

The standards call for learning mathematical content in the context of real-world situations, using

mathematics to solve problems, and developing habits of mind that foster mastery of mathematics

content as well as mathematical understanding. The standards for kindergarten through grade eight

prepare students for higher mathematics. The standards for higher mathematics reflect theknowledge and skills that are necessary to prepare students for college and career and productive

citizenship.

Implementation of the CCSSM will take time and effort, but it also provides a new and exciting

opportunity to ensure that California students are held to the same high expectations in

mathematics as their national and global peers. While California educators have implemented

standards before, the CCSSM require not only rigorous curriculum and instruction but also conceptual

understanding, procedural skill and fluency, and the ability to apply mathematics. In short, the

standards call for meeting the challenges of the 21st

century through innovation.

DR. MICHAEL KIRST, President

California State Board of Education

TOM TORLAKSON

State Superintendent of Public Instruction


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MATHEMATICS |INTRODUCTION


California Department of Education ii |

Introduction

All students need a high-quality mathematics program designed to prepare them to graduate from

high school ready for college and careers. In support of this goal, California adopted the Common

Core State Standards for Mathematics with California Additions (CCSSM) in June 2010, replacing the

1997 statewide mathematics academic standards. As part of the modification of the CCSSM in

January 2013, the California State Board of Education also approved higher mathematics standards

organized into model courses.

The CCSSM are designed to be robust, linked within and across grades, and relevant to the real world,

reflecting the knowledge and skills that our young people need for success in college and careers.

With Californias students fully prepared for the future, our students will be positioned to compete

successfully in the global economy

The development of these standards began as a voluntary, state-led effort coordinated by the Council

of Chief State School Officers (CCSSO) and the National Governors Association Center for Best

Practices (NGA) committed to developing a set of standards that would help prepare students for

success in career and college. The CCSSM are based on evidence of the skills and knowledge needed

for college and career readiness and an expectation that students be able to both know and do

mathematics by solving a range of problems and engaging in key mathematical practices.

The development of these standards was informed by international benchmarking and began with

research-based learning progressions detailing what is known about how students mathematical

knowledge, skills, and understanding develop over time. The progression from kindergarten

standards to standards for higher mathematics exemplifies the three principles of focus, coherence,

and rigor that are the basis for the CCSSM.

The first principle, focus, implies that instruction should focus deeply on only those concepts that are

emphasized in the standards so that students can gain strong foundational conceptual understanding,

a high degree of procedural skill and fluency, and the ability to apply the mathematics they know to

solve problems inside and outside the mathematics classroom. Coherencearises from mathematical

connections. Some of the connections in the standards knit topics together at a single grade level.

Most connections are vertical, as the standards support a progression of increasing knowledge, skill,

and sophistication across the grades. Finally, rigor requires that conceptual understanding,procedural skill and fluency, and application be approached with equal intensity.

Two Types of Standards

The CCSSM include two types of standards: Eight Mathematical Practice Standards (the same at each

grade level) and Mathematical Content Standards (different at each grade level). Together these


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California Department of Education iii |

standards address both habits of mind that students should develop to foster mathematical

understanding and expertise and skills and knowledgewhat students need to know and be able to

do. The mathematical content standards were built on progressions of topics across a number of

grade levels, informed both by research on children's cognitive development and by the logical

structure of mathematics.

The Standards for Mathematical Practice (MP) are the same at each grade level, with the exception

of an additional practice standard included in the California CCSSM for higher mathematics only:

MP3.1:Students build proofs by induction and proofs by contradiction. CA This standard can be

seen as an extension of Mathematical Practice 3, in which students construct viable arguments and

critique the reasoning of others. Ideally, several MP standards will be evident in each lesson as they

interact and overlap with each other. The MP standards are not a checklist; they are the basis for

mathematics instruction and learning. Structuring the MP standards can help educators recognize

opportunities for students to engage with mathematics in grade-appropriate ways. The eight MP

standards can be grouped into the four categories as illustrated in the following chart.

Structuring the Standards for Mathematical Practice1

The CCSSM call for mathematical practices and mathematical content to be connected as students

engage in mathematical tasks. These connections are essential to support the development of

1McCallum, Bill. 2011. Structuring the Mathematical Practices.http://commoncoretools.me/wp-content/uploads/2011/

03/practices.pdf(accessed April 1, 2013).

1.Makesenseofproblemsan

dpersevereinsolving

them.

6.Attendtoprecision.

2. Reason abstractly and

quantitatively.

3. Construct viable arguments and

critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools

strategically.

7. Look for and make use of

structure.

8. Look for and express regularity in

repeated reasoning.

Reasoning and explaining

Modeling and using tools

Seeing structure and generalizing
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California Department of Education iv |

students broader mathematical understanding students who lack understanding of a topic may rely

on procedures too heavily. The MP standards must be taught as carefully and practiced as

intentionally as the Mathematical Content Standards. Neither should be isolated from the other;

effective mathematics instruction occurs when these two halves of the CCSSM come together in a

powerful whole.

How to Read the Standards

KindergartenGrade 8

In kindergarten through grade eight the CCSSM are organized by grade level and then by domains

(clusters of standards that address big ideas and support connections of topics across the grades),

clusters (groups of related standards inside domains) and finally by the standards (what students

should understand and be able to do). The standards do not dictate curriculum or pedagogy. For

example, just because Topic A appears before Topic B in the standards for a given grade, it does notmean that Topic A must be taught before Topic B.

The code for each standard begins with the grade level, followed by the domain code, and the

standard number. For example, 3.NBT 2. would be the second standard in the Number and

Operations in Base Ten domain of the standards for grade three.

Number and Operations in Base Ten 3.NBTUse place value understanding and properties of operations to perform multi-digit

arithmetic.1. Use place value understanding to round whole numbers to the nearest 10 or

100.

2. Fluently add and subtract within 1000 using strategies and algorithms based

on place value, properties of operations, and/or the relationship between

addition and subtraction.

3. Multiply one-digit whole numbers by multiples of 10 in the range 1090 (e.g.,

9 80, 5 60) using strategies based on place value and properties of

operations.

Higher Mathematics

In California, the CCSSM for higher mathematics are organized into both model courses and

conceptual categories. The higher mathematics courses adopted by the State Board of Education in

January 2013 are based on the guidance provided in Appendix A published by the Common Core

Domain

ClusterStandard


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California Department of Education v |

State Standards Initiative.2The model courses for higher mathematics are organized into two

pathways: traditional and integrated. The traditional pathway consists of the higher mathematics

standards organized along more traditional lines into Algebra I, Geometry and Algebra II courses. The

integrated pathway consists of the courses Mathematics I, II and III. The integrated pathway presents

higher mathematics as a connected subject, in that each course contains standards from all six of the

conceptual categories. In addition, two advanced higher mathematics courses were retained from the1997 mathematics standards, Advanced Placement Probability and Statistics and Calculus.

The standards for higher mathematics are also listed in conceptual categories:

Number and Quantity

Algebra

Functions

Modeling

Geometry

Statistics and Probability

The conceptual categories portray a coherent view of higher mathematics based on the realization

that students work on a broad topic, such as functions, crosses a number of traditional course

boundaries. As local school districts develop a full range of courses and curriculum in higher

mathematics, the organization of standards by conceptual categories offers a starting point for

discussing course content.

The code for each higher mathematics standard begins with the identifier for the conceptual category

code (N, A, F, G, S), followed by the domain code, and the standard number. For example, F-LE.5

would be the fifth standard in the Linear, Quadratic, and Exponential Models domain of the

conceptual category of Functions.

Functions

Linear, Quadratic, and Exponential Models F-LEInterpret expressions for functions in terms of the situation they model.

5. Interpret the parameters in a linear or exponential function in terms of a context.

6. Apply quadratic functions to physical problems, such as the motion of an object

under the force of gravity. CA

2Appendix A provides guidance to the field on developing higher mathematics courses. This appendix is available on the

Common Core State Standards Initiative Web site at:http://www.corestandards.org/Math.

Conceptual

Category

Domain

Conceptual

Category and

Domain Codes

ModelingStandard

California Addition: Bold

Font + CA

Cluster

Heading
http://www.corestandards.org/Mathhttp://www.corestandards.org/Mathhttp://www.corestandards.org/Mathhttp://www.corestandards.org/Math


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California Department of Education vi |

The star symbol () following the standard indicates those that are also Modeling standards.

Modeling is best interpreted not as a collection of isolated topics but in relation to other standards.

Making mathematical models is a MP standard and specific modeling standards appear throughout

the higher mathematics standards indicated by a star symbol (). Additional mathematics that

students should learn in order to take advanced courses such as calculus, advanced statistics, or

discrete mathematics is indicated by a (+) symbol. Standards with a (+) symbol may appear in coursesintended for all students.


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California Department of Education 1 |

Mathematics | Standards for

Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics

educators at all levels should seek to develop in their students. These practices rest on importantprocesses and proficiencies with longstanding importance in mathematics education. The first of

these are the NCTM process standards of problem solving, reasoning and proof, communication,

representation, and connections. The second are the strands of mathematical proficiency specified

in the National Research Councils reportAdding It Up: adaptive reasoning, strategic competence,

conceptual understanding (comprehension of mathematical concepts, operations and relations),

procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and

appropriately), and productive disposition (habitual inclination to see mathematics as sensible,

useful, and worthwhile, coupled with a belief in diligence and ones own efficacy).

1 Make sense of problems and persevere in solving them.Mathematically proficient students start by explaining to themselves the meaning of a problem

and looking for entry points to its solution. They analyze givens, constraints, relationships, and

goals. They make conjectures about the form and meaning of the solution and plan a solution

pathway rather than simply jumping into a solution attempt. They consider analogous

problems, and try special cases and simpler forms of the original problem in order to gain

insight into its solution. They monitor and evaluate their progress and change course if

necessary. Older students might, depending on the context of the problem, transform algebraic

expressions or change the viewing window on their graphing calculator to get the information

they need. Mathematically proficient students can explain correspondences between

equations, verbal descriptions, tables, and graphs or draw diagrams of important features and

relationships, graph data, and search for regularity or trends. Younger students might rely on

using concrete objects or pictures to help conceptualize and solve a problem. Mathematically

proficient students check their answers to problems using a different method, and they

continually ask themselves, Does this make sense? They can understand the approaches of

others to solving complex problems and identify correspondences between different

approaches.

2 Reason abstractly and quantitatively.Mathematically proficient students make sense of quantities and their relationships in problem

situations. They bring two complementary abilities to bear on problems involving quantitative

relationships: the ability to decontextualizeto abstract a given situation and represent itsymbolically and manipulate the representing symbols as if they have a life of their own,

without necessarily attending to their referentsand the ability to contextualize, to pause as

needed during the manipulation process in order to probe into the referents for the symbols

involved. Quantitative reasoning entails habits of creating a coherent representation of the

problem at hand; considering the units involved; attending to the meaning of quantities, not


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MATHEMATICS |STANDARDS FOR MATHEMATICAL PRACTICE



just how to compute them; and knowing and flexibly using different properties of operations

and objects.

3 Construct viable arguments and critique the reasoning of

others.Mathematically proficient students understand and use stated assumptions, definitions, and

previously established results in constructing arguments. They make conjectures and build a

logical progression of statements to explore the truth of their conjectures. They are able to

analyze situations by breaking them into cases, and can recognize and use counterexamples.

They justify their conclusions, communicate them to others, and respond to the arguments of

others. They reason inductively about data, making plausible arguments that take into account

the context from which the data arose. Mathematically proficient students are also able to

compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning

from that which is flawed, andif there is a flaw in an argumentexplain what it is.

Elementary students can construct arguments using concrete referents such as objects,

drawings, diagrams, and actions. Such arguments can make sense and be correct, even thoughthey are not generalized or made formal until later grades. Later, students learn to determine

domains to which an argument applies. Students at all grades can listen or read the arguments

of others, decide whether they make sense, and ask useful questions to clarify or improve the

arguments. Students build proofs by induction and proofs by contradiction.CA 3.1(for higher

mathematics only).

4 Model with mathematics.Mathematically proficient students can apply the mathematics they know to solve problems

arising in everyday life, society, and the workplace. In early grades, this might be as simple as

writing an addition equation to describe a situation. In middle grades, a student might applyproportional reasoning to plan a school event or analyze a problem in the community. By high

school, a student might use geometry to solve a design problem or use a function to describe

how one quantity of interest depends on another. Mathematically proficient students who can

apply what they know are comfortable making assumptions and approximations to simplify a

complicated situation, realizing that these may need revision later. They are able to identify

important quantities in a practical situation and map their relationships using such tools as

diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those

relationships mathematically to draw conclusions. They routinely interpret their mathematical

results in the context of the situation and reflect on whether the results make sense, possibly

improving the model if it has not served its purpose.

5 Use appropriate tools strategically.Mathematically proficient students consider the available tools when solving a mathematical

problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a

calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic

geometry software. Proficient students are sufficiently familiar with tools appropriate for their


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grade or course to make sound decisions about when each of these tools might be helpful,

recognizing both the insight to be gained and their limitations. For example, mathematically

proficient high school students analyze graphs of functions and solutions generated using a

graphing calculator. They detect possible errors by strategically using estimation and other

mathematical knowledge. When making mathematical models, they know that technology can

enable them to visualize the results of varying assumptions, explore consequences, andcompare predictions with data. Mathematically proficient students at various grade levels are

able to identify relevant external mathematical resources, such as digital content located on a

website, and use them to pose or solve problems. They are able to use technological tools to

explore and deepen their understanding of concepts.

6 Attend to precision.Mathematically proficient students try to communicate precisely to others. They try to use

clear definitions in discussion with others and in their own reasoning. They state the meaning

of the symbols they choose, including using the equal sign consistently and appropriately. They

are careful about specifying units of measure, and labeling axes to clarify the correspondencewith quantities in a problem. They calculate accurately and efficiently, express numerical

answers with a degree of precision appropriate for the problem context. In the elementary

grades, students give carefully formulated explanations to each other. By the time they reach

high school they have learned to examine claims and make explicit use of definitions.

7 Look for and make use of structure.Mathematically proficient students look closely to discern a pattern or structure. Young

students, for example, might notice that three and seven more is the same amount as seven

and three more, or they may sort a collection of shapes according to how many sides the

shapes have. Later, students will see 7 x 8 equals the well-remembered 7 x 5 + 7 x 3, in

preparation for learning about the distributive property. In the expressionx2+ 9x + 14, older

students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an

existing line in a geometric figure and can use the strategy of drawing an auxiliary line for

solving problems. They also can step back for an overview and shift perspective. They can see

complicated things, such as some algebraic expressions, as single objects or as being composed

of several objects. For example, they can see 53(xy)2as 5 minus a positive number times a

square and use that to realize that its value cannot be more than 5 for any real numbersx and

y.

8 Look for and express regularity in repeated reasoning.Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts. Upper elementary students might notice when dividing 25

by 11 that they are repeating the same calculations over and over again, and conclude they

have a repeating decimal. By paying attention to the calculation of slope as they repeatedly

check whether points are on the line through (1, 2) with slope 3, middle school students might

abstract the equation (y2)/(x1) = 3. Noticing the regularity in the way terms cancel when


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expanding (x1)(x + 1), (x1)(x2

+x + 1), and (x1)(x3+x

2+x + 1) might lead them to the

general formula for the sum of a geometric series. As they work to solve a problem,

mathematically proficient students maintain oversight of the process, while attending to the

details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to theStandards for Mathematical ContentThe Standards for Mathematical Practice describe ways in which developing student

practitioners of the discipline of mathematics increasingly ought to engage with the subject

matter as they grow in mathematical maturity and expertise throughout the elementary,

middle and high school years. Designers of curricula, assessments, and professional

development should all attend to the need to connect the mathematical practices to

mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and

understanding. Expectations that begin with the word understand are often especially good

opportunities to connect the practices to the content. Students who lack understanding of a

topic may rely on procedures too heavily. Without a flexible base from which to work, they

may be less likely to consider analogous problems, represent problems coherently, justify

conclusions, apply the mathematics to practical situations, use technology mindfully to work

with the mathematics, explain the mathematics accurately to other students, step back for an

overview, or deviate from a known procedure to find a shortcut. In short, a lack of

understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are

potential points of intersection between the Standards for Mathematical Content and the

Standards for Mathematical Practice. These points of intersection are intended to be weighted

toward central and generative concepts in the school mathematics curriculum that most merit

the time, resources, innovative energies, and focus necessary to qualitatively improve the

curriculum, instruction, assessment, professional development, and student achievement in

mathematics.


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K8 Standards


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Mathematics | Kindergarten

In Kindergarten, instructional time should focus on two critical areas:(1) representing, relating, and

operating on whole numbers, initially with sets of objects; and (2) describing shapes and space. Morelearning time in Kindergarten should be devoted to number than to other topics.

(1) Students use numbers, including written numerals, to represent quantities and to solve

quantitative problems, such as counting objects in a set; counting out a given number of

objects; comparing sets or numerals; and modeling simple joining and separating

situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 72 = 5.

(Kindergarten students should see addition and subtraction equations, and student writing

of equations in kindergarten is encouraged, but it is not required.) Students choose,

combine, and apply effective strategies for answering quantitative questions, including

quickly recognizing the cardinalities of small sets of objects, counting and producing setsof given sizes, counting the number of objects in combined sets, or counting the number

of objects that remain in a set after some are taken away.

(2) Students describe their physical world using geometric ideas (e.g., shape, orientation,

spatial relations) and vocabulary. They identify, name, and describe basic two-

dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons,

presented in a variety of ways (e.g., with different sizes and orientations), as well as

three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic

shapes and spatial reasoning to model objects in their environment and to construct

more complex shapes.


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Grade K Overview

Counting and Cardinality

Know number names and the count sequence.

Count to tell the number of objects.

Compare numbers.

Operations and Algebraic Thinking

Understand addition as putting together and adding

to, and understand subtraction as taking apart and

taking from.

Number and Operations in Base Ten

Work with numbers 1119 to gain foundations forplace value.

Measurement and Data

Describe and compare measurable attributes.

Classify objects and count the number of objects in

categories.

Geometry

Identify and describe shapes.

Analyze, compare, create, and compose shapes.

Mathematical Practices

1. Make sense of problems and

persevere in solving them.

2. Reason abstractly andquantitatively.

3. Construct viable arguments

and critique the reasoning of

others.


5.Use appropriate tools

strategically.

6.Attend to precision.


structure.

8. Look for and express regularity

in repeated reasoning.

K


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Grade K

Counting and Cardinality K.CCKnow number names and the count sequence.

1. Count to 100 by ones and by tens.

2. Count forward beginning from a given number within the known sequence (instead of having to begin

at 1).3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0

representing a count of no objects).

Count to tell the number of objects.

4. Understand the relationship between numbers and quantities; connect counting to cardinality.

a. When counting objects, say the number names in the standard order, pairing each object with one

and only one number name and each number name with one and only one object.

b. Understand that the last number name said tells the number of objects counted. The number of

objects is the same regardless of their arrangement or the order in which they were counted.

c. Understand that each successive number name refers to a quantity that is one larger.

5. Count to answer how many? questions about as many as 20 things arranged in a line, a rectangulararray, or a circle, or as many as 10 things in a scattered configuration; given a number from 120, count

out that many objects.

Compare numbers.

6. Identify whether the number of objects in one group is greater than, less than, or equal to the number

of objects in another group, e.g., by using matching and counting strategies.1

7. Compare two numbers between 1 and 10 presented as written numerals.

Operations and Algebraic Thinking K.OAUnderstand addition as putting together and adding to, and understand subtraction as taking apart and

taking from.1. Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps),

acting out situations, verbal explanations, expressions, or equations.

2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or

drawings to represent the problem.

3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or

drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by

using objects or drawings, and record the answer with a drawing or equation.

5. Fluently add and subtract within 5.

1Includes groups with up to ten objects.

2Drawings need not show details, but should show the mathematics in the problem.

(This applies wherever drawings are mentioned in the Standards)

K


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Grade K

Number and Operations in Base Ten K.NBTWork with numbers 1119 to gain foundations for place value.

1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using

objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 =

10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six,seven, eight, or nine ones.

Measurement and Data K.MDDescribe and compare measurable attributes.

1. Describe measurable attributes of objects, such as length or weight. Describe several measurable

attributes of a single object.

2. Directly compare two objects with a measurable attribute in common, to see which object has more

of/less of the attribute, and describe the difference. For example, directly compare the heights of

twochildren and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.

3. Classify objects into given categories; count the numbers of objects in each category and sort the

categories by count.3

Geometry K.GIdentify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and

spheres).

1. Describe objects in the environment using names of shapes, and describe the relative positions of these

objects using terms such as above, below, beside, in front of, behind, and next to.

2. Correctly name shapes regardless of their orientations or overall size.

3. Identify shapes as two-dimensional (lying in a plane, flat) or three-dimensional (solid).

Analyze, compare, create, and compose shapes.

4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using

informal language to describe their similarities, differences, parts (e.g., number of sides and

vertices/corners) and other attributes (e.g., having sides of equal length).

5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing

shapes.

6. Compose simple shapes to form larger shapes. For example, Can you join these two triangles with full

sides touching to make a rectangle?

3Limit category counts to be less than or equal to 10.

K


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Mathematics | Grade 1

In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of

addition, subtraction, and strategies for addition and subtraction within 20; (2) developingunderstanding of whole number relationships and place value, including grouping in tens and ones;

(3) developing understanding of linear measurement and measuring lengths as iterating length units;

and (4) reasoning about attributes of, and composing and decomposing geometric shapes.

(1) Students develop strategies for adding and subtracting whole numbers based on their

prior work with small numbers. They use a variety of models, including discrete objects

and length-based models (e.g., cubes connected to form lengths), to model add-to, take-

from, put-together, take-apart, and compare situations to develop meaning for the

operations of addition and subtraction, and to develop strategies to solve arithmetic

problems with these operations. Students understand connections between counting and

addition and subtraction (e.g., adding two is the same as counting on two). They use

properties of addition to add whole numbers and to create and use increasingly

sophisticated strategies based on these properties (e.g., making tens) to solve addition

and subtraction problems within 20. By comparing a variety of solution strategies, children

build their understanding of the relationship between addition and subtraction.

(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to add

within 100 and subtract multiples of 10. They compare whole numbers (at least to 100) to

develop understanding of and solve problems involving their relative sizes. They think of

whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the

numbers 11 to 19 as composed of a ten and some ones). Through activities that buildnumber sense, they understand the order of the counting numbers and their relative

magnitudes.

(3) Students develop an understanding of the meaning and processes of measurement,

including underlying concepts such as iterating (the mental activity of building up the

length of an object with equal-sized units) and the transitivity principle for indirect

measurement.1

(4) Students compose and decompose plane or solid figures (e.g., put two triangles together

to make a quadrilateral) and build understanding of part-whole relationships as well as the

properties of the original and composite shapes. As they combine shapes, they recognize

them from different perspectives and orientations, describe their geometric attributes,

and determine how they are alike and different, to develop the background for

measurement and for initial understandings of properties such as congruence and

symmetry.

1Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use

this technical term.


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Grade 1 Overview


Represent and solve problems involving addition and

subtraction.

Understand and apply properties of operations and the

relationship between addition and subtraction.

Add and subtract within 20.

Work with addition and subtraction equations.


Extend the counting sequence.

Understand place value.

Use place value understanding and properties of

operations to add and subtract.


Measure lengths indirectly and by iterating length units.

Tell and write time.

Represent and interpret data.

Geometry Reason with shapes and their attributes.

1



persevere in solving them.2. Reason abstractly and

quantitatively.



others.

4.Model with mathematics.


strategically.



structure.




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Grade 1

Operations and Algebraic Thinking 1.OARepresent and solve problems involving addition and subtraction.

1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking

from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using

objects, drawings, and equations with a symbol for the unknown number to represent the problem.

2

2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to

20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent

the problem.

Understand and apply properties of operations and the relationship between addition and subtraction.

3. Apply properties of operations as strategies to add and subtract.3Examples: If 8 + 3 = 11 is known, then

3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers

can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

4. Understand subtraction as an unknown-addend problem. For example, subtract 10 8 by finding the

number that makes 10 when added to 8.


5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use

strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number

leading to a ten (e.g., 134 = 1331 = 101 = 9); using the relationship between addition and

subtraction (e.g., knowing that 8 + 4 = 12, one knows 128 = 4); and creating equivalent but easier or

known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Work with addition and subtraction equations.

7. Understand the meaning of the equal sign, and determine if equations involving addition and

subtraction are true or false. For example, whichof the following equations are true and which are

false? 6 = 6, 7 = 8 1,5 + 2 = 2 + 5, 4 + 1 = 5 + 2.8. Determine the unknown whole number in an addition or subtraction equation relating three whole

numbers. For example, determine theunknown number that makes the equation true in each of the

equations 8 +? = 11, 5 = 3, 6 + 6 = .

Number and Operations in Base Ten 1.NBTExtend the counting sequence.

1. Count to 120, starting at any number less than 120. In this range, read and write numerals and

represent a number of objects with a written numeral.


2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understandthe following as special cases:

a. 10 can be thought of as a bundle of ten ones called a ten.

2See Glossary, Table 1.

3Students need not use formal terms for these properties.

1


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Grade 1

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,

eight, or nine ones.

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,

eight, or nine tens (and 0 ones).

3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results ofcomparisons with the symbols >, =, and


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Grade 1

3. Partition circles and rectangles into two and four equal shares, describe the shares using the words

halves,fourths, and quarters, and use the phrases half of,fourth of, and quarter of. Describe the whole

as two of, or four of the shares. Understand for these examples that decomposing into more equal

shares creates smaller shares.

1


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In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-

ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure;

and (4) describing and analyzing shapes.

(1) Students extend their understanding of the base-ten system. This includes ideas of

counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number

relationships involving these units, including comparing. Students understand multi-digit

numbers (up to 1000) written in base-ten notation, recognizing that the digits in each

place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5

tens + 3 ones).

(2) Students use their understanding of addition to develop fluency with addition and

subtraction within 100. They solve problems within 1000 by applying their understanding

of models for addition and subtraction, and they develop, discuss, and use efficient,

accurate, and generalizable methods to compute sums and differences of whole numbers

in base-ten notation, using their understanding of place value and the properties of

operations. They select and accurately apply methods that are appropriate for the context

and the numbers involved to mentally calculate sums and differences for numbers with

only tens or only hundreds.

(3) Students recognize the need for standard units of measure (centimeter and inch) and they

use rulers and other measurement tools with the understanding that linear measure

involves an iteration of units. They recognize that the smaller the unit, the more iterations

they need to cover a given length.

(4) Students describe and analyze shapes by examining their sides and angles. Students

investigate, describe, and reason about decomposing and combining shapes to make

other shapes. Through building, drawing, and analyzing two- and three-dimensional

shapes, students develop a foundation for understanding area, volume, congruence,

similarity, and symmetry in later grades.


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Grade 2 Overview


Represent and solve problems involving addition

and subtraction.


Work with equal groups of objects to gain

foundations for multiplication.




operations to add and subtract.


Measure and estimate lengths in standard units.

Relate addition and subtraction to length.

Work with time and money.


Geometry

Reason with shapes and their attributes.

2





quantitatively.



others.



strategically.

6.Attend to precision.7. Look for and make use of

structure.




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Grade 2

Operations and Algebraic Thinking 2.OARepresent and solve problems involving addition and subtraction.

1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations

of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions,

e.g., by using drawings and equations with a symbol for the unknown number to represent theproblem.1


2. Fluently add and subtract within 20 using mental strategies.2By end of Grade 2, know from memory all

sums of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.

3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by

pairing objects or counting them by 2s; write an equation to express an even number as a sum of two

equal addends.

4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and upto 5 columns; write an equation to express the total as a sum of equal addends.

Number and Operations in Base Ten 2.NBTUnderstand place value.

1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and

ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens called a hundred.

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six,

seven, eight, or nine hundreds (and 0 tens and 0 ones).

2. Count within 1000; skip-count by 2s, 5s, 10s, and 100s. CA

3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =,

and < symbols to record the results of comparisons.

Use place value understanding and properties of operations to add and subtract.

5. Fluently add and subtract within 100 using strategies based on place value, properties of operations,

and/or the relationship between addition and subtraction.

6. Add up to four two-digit numbers using strategies based on place value and properties of operations.

7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value,

properties of operations, and/or the relationship between addition and subtraction; relate the strategy

to a written method. Understand that in adding or subtracting three-digit numbers, one adds or

subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary tocompose or decompose tens or hundreds.

7.1 Use estimation strategies to make reasonable estimates in problem solving. CA

8. Mentally add 10 or 100 to a given number 100900, and mentally subtract 10 or 100 from a given

number 100900.


2See standard 1.OA.6 for a list of mental strategies.

2


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Grade 2

9. Explain why addition and subtraction strategies work, using place value and the properties of

operations.3

Measurement and Data 2.MDMeasure and estimate lengths in standard units.

1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks,

meter sticks, and measuring tapes.

2. Measure the length of an object twice, using length units of different lengths for the two

measurements; describe how the two measurements relate to the size of the unit chosen.

3. Estimate lengths using units of inches, feet, centimeters, and meters.

4. Measure to determine how much longer one object is than another, expressing the length difference in

terms of a standard length unit.

Relate addition and subtraction to length.

5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in the

same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for theunknown number to represent the problem.

6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points

corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100

on a number line diagram.

Work with time and money.

7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Know

relationships of time (e.g., minutes in an hour, days in a month, weeks in a year). CA

8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols

appropriately. Example: If you have 2dimes and 3 pennies, how many cents do you have?


9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by

making repeated measurements of the same object. Show the measurements by making a line plot,

where the horizontal scale is marked off in whole-number units.

10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four

categories. Solve simple put-together, take-apart, and compare problems4using information presented

in a bar graph.

Geometry 2.GReason with shapes and their attributes.

1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given

number of equal faces.5Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of

them.

3Explanations may be supported by drawings or objects.


5Sizes are compared directly or visually, not compared by measuring.

2


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Grade 2

3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words

halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.

Recognize that equal shares of identical wholes need not have the same shape.

2


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In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of

multiplication and division and strategies for multiplication and division within 100; (2) developing

understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing

understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing

two-dimensional shapes.

(1) Students develop an understanding of the meanings of multiplication and division of

whole numbers through activities and problems involving equal-sized groups, arrays, and

area models; multiplication is finding an unknown product, and division is finding an

unknown factor in these situations. For equal-sized group situations, division can require

finding the unknown number of groups or the unknown group size. Students use

properties of operations to calculate products of whole numbers, using increasinglysophisticated strategies based on these properties to solve multiplication and division

problems involving single-digit factors. By comparing a variety of solution strategies,

students learn the relationship between multiplication and division.

(2) Students develop an understanding of fractions, beginning with unit fractions. Students

view fractions in general as being built out of unit fractions, and they use fractions along

with visual fraction models to represent parts of a whole. Students understand that the

size of a fractional part is relative to the size of the whole. For example,1/2 of the paint in

a small bucket could be less paint than1/3 of the paint in a larger bucket, but 1/3 of a

ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3

equal parts, the parts are longer than when the ribbon is divided into 5 equal parts.

Students are able to use fractions to represent numbers equal to, less than, and greater

than one. They solve problems that involve comparing fractions by using visual fraction

models and strategies based on noticing equal numerators or denominators.

(3) Students recognize area as an attribute of two-dimensional regions. They measure the

area of a shape by finding the total number of same-size units of area required to cover

the shape without gaps or overlaps, a square with sides of unit length being the standard

unit for measuring area. Students understand that rectangular arrays can be decomposed

into identical rows or into identical columns. By decomposing rectangles into rectangular

arrays of squares, students connect area to multiplication, and justify using multiplicationto determine the area of a rectangle.

(4) Students describe, analyze, and compare properties of two-dimensional shapes. They

compare and classify shapes by their sides and angles, and connect these with definitions

of shapes. Students also relate their fraction work to geometry by expressing the area of

part of a shape as a unit fraction of the whole.


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Grade 3 Overview


Represent and solve problems involving

multiplication and division.

Understand properties of multiplication and therelationship between multiplication and division.

Multiply and divide within 100.

Solve problems involving the four operations, and

identify and explain patterns in arithmetic.



operations to perform multi-digit arithmetic.

Number and OperationsFractions

Develop understanding of fractions as numbers.


Solve problems involving measurement and estimation of intervals of time, liquid volumes,

and masses of objects.


Geometric measurement: understand concepts of area and relate area to multiplication and

to addition.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish

between linear and area measures.

Geometry

Reason with shapes and their attributes.

3




2. Reason abstractly andquantitatively.



others.



strategically.



structure.




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Grade 3

Operations and Algebraic Thinking 3.OARepresent and solve problems involving multiplication and division.

1. Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of

7 objects each. For example, describea context in which a total number of objects can be expressed as 5

7.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 8 as the number of objects in

each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56

objects are partitioned into equal shares of 8 objects each. Forexample, describe a context in which a

number of shares or a number ofgroups can be expressed as 568.

3. Use multiplication and division within 100 to solve word problems in situations involving equal groups,

arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the

unknown number to represent the problem.1

4. Determine the unknown whole number in a multiplication or division equation relating three whole

numbers. For example, determine the unknown number that makes the equation true in each of the

equations 8 ? = 48, 5 = 3, 6 6 = ?.

Understand properties of multiplication and the relationship between multiplication and division.

5. Apply properties of operations as strategies to multiply and divide.2Examples: If 6 4 = 24 is known,

then 4 6 = 24 is also known.(Commutative property of multiplication.) 3 5 2 can be found by 3 5 =

15, then 15 2 = 30, or by 5 2 = 10, then 3 10 = 30. (Associativeproperty of multiplication.) Knowing

that 8 5 = 40 and 8 2 = 16, one can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56.

(Distributiveproperty.)

6. Understand division as an unknown-factor problem. For example, find 32 8 by finding the number that

makes 32 when multiplied by 8.

Multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication

and division (e.g., knowing that 8 5 = 40, one knows 40 5 = 8) or properties of operations. By the end

of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

8. Solve two-step word problems using the four operations. Represent these problems using equations

with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental

computation and estimation strategies including rounding.3

9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain

them using properties of operations. For example, observe that 4 timesa number is always even, and

explain why 4 times a number can be decomposed into two equal addends.


2Students need not use formal terms for these properties.

3This standard is limited to problems posed with whole numbers and having whole-number answers; students should

know how to perform operations in the conventional order when there are no parentheses to specify a particular order

(Order of Operations).

3


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Grade 3

Number and Operations in Base Ten 3.NBTUse place value understanding and properties of operations to perform multi-digit arithmetic.4

1. Use place value understanding to round whole numbers to the nearest 10 or 100.

2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties

of operations, and/or the relationship between addition and subtraction.3. Multiply one-digit whole numbers by multiples of 10 in the range 1090 (e.g., 9 80, 5 60) using

strategies based on place value and properties of operations.

Number and OperationsFractions5 3.NFDevelop understanding of fractions as numbers.

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal

parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the

whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the

endpoint of the part based at 0 locates the number 1/b on the number line.b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.

Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b

on the number line.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a

number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the

fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole

numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the

same point of a number line diagram.d. Compare two fractions with the same numerator or the same denominator by reasoning about

their size. Recognize that comparisons are valid only when the two fractions refer to the same

whole. Record the results of comparisons with the symbols >, =, or


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Grade 3

involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker

with a measurement scale) to represent the problem.7


3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.Solve one- and two-step how many more and how many less problems using information presented

in scaled bar graphs. For example, draw a bar graph in which each square inthe bar graph might

represent 5 pets.

4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an

inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate

unitswhole numbers, halves, or quarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

5. Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called a unit square, is said to have one square unit of area,

and can be used to measure area.

b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to havean area of n square units.

6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised

units).

7. Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area

is the same as would be found by multiplying the side lengths.

b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context

of solving real world and mathematical problems, and represent whole-number products as

rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side

lengths a and b + c is the sum of a b and a c. Use area models to represent the distributiveproperty in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-

overlapping rectangles and adding the areas of the non-overlapping parts, applying this

technique to solve real world problems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between

linear and area measures.

8. Solve real world and mathematical problems involving perimeters of polygons, including finding the

perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the

same perimeter and different areas or with the same area and different perimeters.

7Excludes multiplicative comparison problems (problems involving notions of times as much; see Glossary, Table 2).

3


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Grade 3

Geometry 3.GReason with shapes and their attributes.

1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share

attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g.,

quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and drawexamples of quadrilaterals that do not belong to any of these subcategories.

2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the

whole. For example, partition a shape into 4parts with equal area, and describe the area of each part as

1/4 of the areaof the shape.

3


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In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and

fluency with multi-digit multiplication, and developing understanding of dividing to find quotients

involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and

subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3)

understanding that geometric figures can be analyzed and classified based on their properties, such

as having parallel sides, perpendicular sides, particular angle measures, and symmetry.

(1) Students generalize their understanding of place value to 1,000,000, understanding the

relative sizes of numbers in each place. They apply their understanding of models for

multiplication (equal-sized groups, arrays, area models), place value, and properties of

operations, in particular the distributive property, as they develop, discuss, and use

efficient, accurate, and generalizable methods to compute products of multi-digit wholenumbers. Depending on the numbers and the context, they select and accurately apply

appropriate methods to estimate or mentally calculate products. They develop fluency

with efficient procedures for multiplying whole numbers; understand and explain why the

procedures work based on place value and properties of operations; and use them to

solve problems. Students apply their understanding of models for division, place value,

properties of operations, and the relationship of division to multiplication as they develop,

discuss, and use efficient, accurate, and generalizable procedures to find quotients

involving multi-digit dividends. They select and accurately apply appropriate methods to

estimate and mentally calculate quotients, and interpret remainders based upon the

context.

(2) Students develop understanding of fraction equivalence and operations with fractions.

They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they

develop methods for generating and recognizing equivalent fractions. Students extend

previous understandings about how fractions are built from unit fractions, composing

fractions from unit fractions, decomposing fractions into unit fractions, and using the

meaning of fractions and the meaning of multiplication to multiply a fraction by a whole

number.

(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through

building, drawing, and analyzing two-dimensional shapes, students deepen theirunderstanding of properties of two-dimensional objects and the use of them to solve

problems involving symmetry.


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Grade 4 Overview


Use the four operations with whole numbers to

solve problems.

Gain familiarity with factors and multiples.

Generate and analyze patterns.


Generalize place value understanding for multi-digit

whole numbers.


operations to perform multi-digit arithmetic.

Number and OperationsFractions

Extend understanding of fraction equivalence and

ordering.

Build fractions from unit fractions by applying and

extending previous understandings of operations on

whole numbers.

Understand decimal notation for fractions, and compare decimal fractions.

Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to

a smaller unit.


Geometric measurement: understand concepts of angle and measure angles.

Geometry

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

4





quantitatively.



others.



strategically.

6.Attend to precision.7. Look for and make use of

structure.




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Grade 4

Operations and Algebraic Thinking 4.OAUse the four operations with whole numbers to solve problems.

1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5

times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicativecomparisons as multiplication equations.

2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings

and equations with a symbol for the unknown number to represent the problem, distinguishing

multiplicative comparison from additive comparison.1

3. Solve multistep word problems posed with whole numbers and having whole-number answers using

the four operations, including problems in which remainders must be interpreted. Represent these

problems using equations with a letter standing for the unknown quantity. Assess the reasonableness

of answers using mental computation and estimation strategies including rounding.

Gain familiarity with factors and multiples.

4. Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a

multiple of each of its factors. Determine whether a given whole number in the range 1100 is amultiple of a given one-digit number. Determine whether a given whole number in the range 1100 is

prime or composite.

Generate and analyze patterns.

5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern

that were not explicit in the rule itself. For example, given the rule Add 3 and the starting number 1,

generateterms in the resulting sequence and observe that the terms appear toalternate between odd

and even numbers. Explain informally why thenumbers will continue to alternate in this way.

Number and Operations in Base Ten2 4.NBT

Generalize place value understanding for multi-digit whole numbers.1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it

represents in the place to its right. Forexample, recognize that 700 70 = 10 by applying concepts of

place valueand division.

2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded

form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and , =, or 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to

the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one

way, recording each decomposition by an equation. Justify decompositions, e.g., by using a

visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ;3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 =

8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number

with an equivalent fraction, and/or by using properties of operations and the relationship

between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the samewhole and having like denominators, e.g., by using visual fraction models and equations to

represent the problem.

4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to

represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a

fraction by a whole number. Forexample, use a visual fraction model to express 3 (2/5) as 6

(1/5),recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using

visual fraction models and equations to represent the problem. For example, if each person at a

party willeat 3/8 of a pound of roast beef, and there will be 5 people at the party, how manypounds of roast beef will be needed? Between what two whole numbers does your answer lie?

3Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

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Grade 4

Understand decimal notation for fractions, and compare decimal fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this

technique to add two fractions with respective denominators 10 and 100.4For example, express 3/10 as

30/100, and add 3/10 + 4/100 = 34/100.

6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100;describe a length as 0.62 meters; locate 0.62 on a number line diagram.

7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are

valid only when the two decimals refer to the same whole. Record the results of comparisons with the

symbols >, =, or


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6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the

angle measure of the whole is the sum of the angle measures of the parts. Solve addition and

subtraction problems to find unknown angles on a diagram in real world and mathematical problems,

e.g., by using an equation with a symbol for the unknown angle measure.

Geometry 4.GDraw and identify lines and angles, and classify shapes by properties of their lines and angles.

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel

lines. Identify these in two-dimensional figures.

2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or

the presence or absence of angles of a specified size. Recognize right triangles as a category, and

identify right triangles. (Two dimensional shapes should include special triangles, e.g., equilateral,

isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram,

trapezoid.) CA

3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that thefigure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of

symmetry.

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In Grade 5, instructional time should focus o

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Ccs s Math April 2013